N-growing hierarchy

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The N-growing hierarchy is a hierarchy/notation based on the fast-growing hierarchy, created by Japanese googologist Aeton (2013) ^[1].

Definition[]

\([m]_0(n) = m\times n\)
\([m]_{\alpha+1}(n) = [m]^{n-1}_\alpha(m)\), and if \(n=1\), \([m]_{\alpha+1}(1)=[m]^0_\alpha(m)=m\)
\([m]_\alpha(n) = [m]_{\alpha[n]}(m)\), when \(\alpha\) is a limit ordinal and \(\alpha[n]\) is the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\).

And when \(m=10\), it can be called 10-growing hierarchy. And similarly, 3-growing hierarchy, 16-growing hierarchy, or Googol-growing hierarchy are also possible.

However, If \(m=n\), it is called Diagonal n-growing hierarchy and its notation changes as follows.

\((N_\alpha(n) = [n]_\alpha(n))\)
\(N_0(n) = n\times n=n^2\)
\(N_{\alpha+1}(n) = N^{n-1}_\alpha(n)\)
\(N_\alpha(n) = N_{\alpha[n]}(n)\)

Examples[]

This function is exactly equal to up-arrow notation, and probably array notation, but for that reason, when \(m=2\) and \(\alpha\geq\omega\), it does not grow well.

\([16]_4(8) = 16\uparrow^4 8\)
\([10]_{\omega+1}(100) = \{10,100,1,2\}=\) Corporal
\([3]^{64}_{\omega}(4)\) = Graham's number \(\lesssim[4]_{\omega+1}(65) = \{4,65,1,2\}\)
\([4]_{\omega^2+1}(64) = \{4,64,1,1,2\}<\) Fish number 1

\(N_\omega(3) = [3]_3(3) = 3\uparrow^3 3=\) Tritri
\(N_{\omega^2}(10) = \{10,10,10,10\}=\) General

Because of the reason that \([m]_{\omega^\omega}(n)=\{m,n+2(1)2\}\), this function doesn't match exactly over \(\{m,n(1)2\}\) level of BEAF, in \(\alpha\geq\omega^\omega\) level.

\(N_{\omega^{98}}(10) = [10]_{\omega^\omega}(98) = \{10,100 (1) 2\}=\) Goobol
\([10]_{\omega^\omega}([10]_{\omega^\omega}(98)-2)=\) goobolplex \(\approx[10]^2_{\omega^\omega}(98)\)

Sources[]

↑ n-growing hierarchy (Japanese Page)

See also[]

Googology in Japan

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea

Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system

Implementation of existing works into programs

By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation